Where is $query set? The line above is executing the query, without capturing the result. Looks like it should be replaced with:
$query = " UPDATE Liberty SET firstname='$FName' , lastname='$LName' , PatientPhone='$PatientPhone' , FacName='$FacName' , FacAddress='$FacAddress' , PatRoom='$PatRoom' , FacCity='$FacCity' , FacState='$FacState' , FacZip='$FacZip' , FacPhone='$FacPhone' , CaregiverName='$CaregiverName' , CaregiverContactPhone='$CaregiverContactPhone' , MedDispenseTime1='$MedDispenseTime1' , MedDispenseTime2='$MedDispenseTime2' , MedDispenseTime3='$MedDispenseTime3' , MedDispenseTime4='$MedDispenseTime4' , MedDispenseTime5='$MedDispenseTime5' , MedDispenseTime6='$MedDispenseTime6' , Month='$Month' , Year='$Year' WHERE PatientID='$id";